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  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "oXMSF1dYc3jK"
   },
   "source": [
    "# Residual Networks (ResNet)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "6u-0RfV4c3jL"
   },
   "source": [
    "As we design increasingly deeper networks it becomes imperative to understand how adding layers can\n",
    "increase the complexity and expressiveness of the network. Even more important is the ability to design\n",
    "networks where adding layers makes networks strictly more expressive rather than just different. To make\n",
    "some progress we need a bit of theory."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "YpAXoD5Qc3jM"
   },
   "source": [
    "## Function Classes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "TaakpHrZc3jN"
   },
   "source": [
    "Consider F, the class of functions that a specific network architecture (together with learning rates and\n",
    "other hyperparameter settings) can reach. That is, for all f ∈ F there exists some set of parameters W that\n",
    "can be obtained through training on a suitable dataset. Let’s assume that f∗ is the function that we really\n",
    "would like to find. If it’s in F, we’re in good shape but typically we won’t be quite so lucky. Instead, we\n",
    "will try to find some fF∗ which is our best bet within F. For instance, we might try finding it by solving the\n",
    "following optimization problem:\n",
    "                     \n",
    "                     fF∗ := argmin L(X,Y,f) subject to f ∈ F \n",
    "\n",
    "It is only reasonable to assume that if we design a different and more powerful architecture F′ we should\n",
    "arrive at a better outcome. In other words, we would expect that fF∗′ is ‘better’ than fF∗ . However, if F ̸⊆ F′\n",
    "there is no guarantee that this should even happen. In fact, fF∗′ might well be worse. This is a situation\n",
    "that we often encounter in practice - adding layers doesn’t only make the network more expressive, it also\n",
    "changes it in sometimes not quite so predictable ways. The picture below illustrates this in slightly abstract\n",
    "terms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "IwKt22zNc3jO",
    "outputId": "0e17aa01-af5f-4e66-f55a-62cd5752328e"
   },
   "outputs": [
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      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import SVG\n",
    "SVG(filename=\"../img/functionclasses.svg\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "4dWl0G66c3jU"
   },
   "source": [
    "Only if larger function classes contain the smaller ones are we guaranteed that increasing them strictly\n",
    "increases the expressive power of the network. This is the question that He et al, 2016 considered when\n",
    "working on very deep computer vision models. At the heart of ResNet is the idea that every additional layer\n",
    "should contain the identity function as one of its elements. This means that if we can train the newly-added\n",
    "layer into an identity mapping $f (x) = x$, the new model will be as effective as the original model. As the\n",
    "new model may get a better solution to fit the training data set, the added layer might make it easier to\n",
    "reduce training errors. Even better, the identity function rather than the null $f (x) = 0$ should be the the\n",
    "simplest function within a layer.\n",
    "These considerations are rather profound but they led to a surprisingly simple solution, a residual block.\n",
    "With it, He et al, $2015^{146}$ won the ImageNet Visual Recognition Challenge in 2015. The design had a\n",
    "profound influence on how to build deep neural networks."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "_P_L3ZJ6c3jV"
   },
   "source": [
    "## Residual Blocks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "cAET27QDc3jW"
   },
   "source": [
    "Let us focus on a local neural network, as depicted below. Denote the input by $x$. We assume that the ideal\n",
    "mapping we want to obtain by learning is $f (x)$, to be used as the input to the activation function. The\n",
    "portion within the dotted-line box in the left image must directly fit the mapping $f (x)$. This can be tricky\n",
    "if we don’t need that particular layer and we would much rather retain the input $x$. The portion within the\n",
    "dotted-line box in the right image now only needs to parametrize the deviation from the identity, since we\n",
    "return $x + f (x)$. In practice, the residual mapping is often easier to optimize. We only need to set $f (x) = 0$.\n",
    "The right image in the figure below illustrates the basic Residual Block of ResNet. Similar architectures\n",
    "were later proposed for sequence models which we will study later."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "m4jR5sYTc3jX",
    "outputId": "b54cb6e1-6abd-4c08-e894-57de32c273fa"
   },
   "outputs": [
    {
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      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SVG(filename=\"../img/residual-block.svg\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "oxpww5yOc3jc"
   },
   "source": [
    "ResNet follows VGG’s full 3 × 3 convolutional layer design. The residual block has two 3 × 3 convolutional\n",
    "layers with the same number of output channels. Each convolutional layer is followed by a batch normal-\n",
    "ization layer and a ReLU activation function. Then, we skip these two convolution operations and add the\n",
    "input directly before the final ReLU activation function. This kind of design requires that the output of the\n",
    "two convolutional layers be of the same shape as the input, so that they can be added together. If we want\n",
    "to change the number of channels or the the stride, we need to introduce an additional 1 × 1 convolutional\n",
    "layer to transform the input into the desired shape for the addition operation. Let’s have a look at the code\n",
    "below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "1paetVAl8vP0"
   },
   "outputs": [],
   "source": [
    "import sys\n",
    "sys.path.insert(0, '..')\n",
    "\n",
    "import d2l\n",
    "import torch\n",
    "import torch.nn as nn\n",
    "import torch.nn.functional as F\n",
    "\n",
    "# This class has been saved in the d2l package for future use\n",
    "class Residual(nn.Module):\n",
    "  \n",
    "  def __init__(self,input_channels, num_channels, use_1x1conv=False, strides=1, **kwargs):\n",
    "    super(Residual, self).__init__(**kwargs)\n",
    "    self.conv1 = nn.Conv2d(input_channels, num_channels,kernel_size=3, padding=1, stride=strides)\n",
    "    self.conv2 = nn.Conv2d(num_channels, num_channels, kernel_size=3, padding=1)\n",
    "    if use_1x1conv:\n",
    "      self.conv3 = nn.Conv2d(input_channels, num_channels, kernel_size=1, stride=strides)\n",
    "    else:\n",
    "      self.conv3 = None\n",
    "    self.bn1 = nn.BatchNorm2d(num_channels)\n",
    "    self.bn2 = nn.BatchNorm2d(num_channels)\n",
    "    self.relu = nn.ReLU(inplace=True)\n",
    "  \n",
    "  def forward(self, X):\n",
    "    \n",
    "    Y = self.relu(self.bn1(self.conv1(X)))\n",
    "    Y = self.bn2(self.conv2(Y))\n",
    "    if self.conv3:\n",
    "      X = self.conv3(X)\n",
    "    Y += X\n",
    "    Y =self.relu(Y)\n",
    "    return Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "guFKBg8Hc3jf"
   },
   "source": [
    "This code generates two types of networks: one where we add the input to the output before applying the\n",
    "ReLU nonlinearity, and whenever use_1x1conv=True, one where we adjust channels and resolution by means\n",
    "of a 1 × 1 convolution before adding. The diagram below illustrates this:\n",
    "\n",
    "Now let us look at a situation where the input and output are of the same shape."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 34
    },
    "colab_type": "code",
    "id": "6brP4dJzXF-m",
    "outputId": "4e61ad06-c2ab-4006-993d-740495e31e9a"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([4, 3, 6, 6])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "blk = Residual(3,3)\n",
    "X = torch.rand(4, 3, 6, 6)\n",
    "Y = blk(X)\n",
    "Y.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "GBV6vyqic3jl"
   },
   "source": [
    "We also have the option to halve the output height and width while increasing the number of output channels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 34
    },
    "colab_type": "code",
    "id": "mu_vVqDZXskn",
    "outputId": "b6c8102c-6f9a-41d5-ef5f-30925a978f22"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([4, 6, 3, 3])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "blk = Residual(3,6, use_1x1conv=True, strides=2)\n",
    "blk(X).shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "BAjUmDO1c3jq",
    "outputId": "c4085cea-7568-414d-f9ba-0596166047cf"
   },
   "outputs": [
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SVG(filename=\"../img/ResNetBlock.svg\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ER88Q6Yec3ju"
   },
   "source": [
    "## ResNet Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "m1QUNit8c3jv"
   },
   "source": [
    "The first two layers of ResNet are the same as those of the GoogLeNet we described before: the 7 × 7\n",
    "convolutional layer with 64 output channels and a stride of 2 is followed by the 3 × 3 maximum pooling layer\n",
    "with a stride of 2. The difference is the batch normalization layer added after each convolutional layer in\n",
    "ResNet."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "kXxfNwINa7YT"
   },
   "outputs": [],
   "source": [
    "b1 = nn.Sequential(nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),\n",
    "                    nn.BatchNorm2d(64),\n",
    "                    nn.ReLU(),\n",
    "                    nn.MaxPool2d(kernel_size=3, stride=2, padding=1))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "VJF6rRMrc3jz"
   },
   "source": [
    "GoogLeNet uses four blocks made up of Inception blocks. However, ResNet uses four modules made up of\n",
    "residual blocks, each of which uses several residual blocks with the same number of output channels. The\n",
    "number of channels in the first module is the same as the number of input channels. Since a maximum\n",
    "pooling layer with a stride of 2 has already been used, it is not necessary to reduce the height and width.\n",
    "In the first residual block for each of the subsequent modules, the number of channels is doubled compared\n",
    "with that of the previous module, and the height and width are halved.\n",
    "\n",
    "Now, we implement this module. Note that special processing has been performed on the first module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "DWhbp4XfeWro"
   },
   "outputs": [],
   "source": [
    "def resnet_block(input_channels, num_channels, num_residuals, first_block=False):\n",
    "  blk = []\n",
    "  for i in range(num_residuals):\n",
    "    if i == 0 and not first_block:\n",
    "      blk.append(Residual(input_channels, num_channels, use_1x1conv=True, strides=2))\n",
    "    else:\n",
    "      blk.append(Residual(num_channels, num_channels))\n",
    "  return blk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ceR5RWgQc3j3"
   },
   "source": [
    "Then, we add all the residual blocks to ResNet. Here, two residual blocks are used for each module. Just like GoogLeNet, we add a global average pooling layer, followed by the fully connected layer\n",
    "output."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "APOwROqqoaa3"
   },
   "outputs": [],
   "source": [
    "class Flatten(nn.Module):\n",
    "    def forward(self, input):\n",
    "        return input.view(input.size(0), -1)\n",
    "b2=nn.Sequential(*resnet_block(64,64,2,first_block=True))\n",
    "b3=nn.Sequential(*resnet_block(64,128,2))\n",
    "b4=nn.Sequential(*resnet_block(128,256,2))\n",
    "b5=nn.Sequential(*resnet_block(256,512,2))\n",
    "net=nn.Sequential(b1,b2,b3,b4,b5,nn.AdaptiveMaxPool2d((1,1)),Flatten(),nn.Linear(512, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "blA0ySzIc3j8"
   },
   "source": [
    "There are 4 convolutional layers in each module (excluding the 1 × 1 convolutional layer). Together with\n",
    "the first convolutional layer and the final fully connected layer, there are 18 layers in total. Therefore, this\n",
    "model is commonly known as ResNet-18. By configuring different numbers of channels and residual blocks\n",
    "in the module, we can create different ResNet models, such as the deeper 152-layer ResNet-152. Although\n",
    "the main architecture of ResNet is similar to that of GoogLeNet, ResNet’s structure is simpler and easier to\n",
    "modify. All these factors have resulted in the rapid and widespread use of ResNet. Below is a diagram of\n",
    "the full ResNet-18."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "E_51ppHjc3j9"
   },
   "source": [
    "## Data Acquisition and Training "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "vMVTsDDpc3j-"
   },
   "source": [
    "We train ResNet on the Fashion-MNIST data set, just like before. The only thing that has changed is the\n",
    "learning rate that decreased again, due to the more complex architecture."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "pnLNxcM-c3j_",
    "outputId": "0edfbebf-8873-4871-b3ea-10bf3caa69ec"
   },
   "outputs": [
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    "SVG(filename=\"../img/ResNetFull.svg\")"
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  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 119
    },
    "colab_type": "code",
    "id": "2eO9Cad8c3kD",
    "outputId": "20a37cc4-4606-4e82-e364-af83336ac632"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "training on cuda:0\n",
      "epoch 1, loss 0.0031, train acc 0.795, test acc 0.618, time 52.1 sec\n",
      "epoch 2, loss 0.0011, train acc 0.891, test acc 0.875, time 51.0 sec\n",
      "epoch 3, loss 0.0009, train acc 0.914, test acc 0.883, time 51.4 sec\n",
      "epoch 4, loss 0.0007, train acc 0.932, test acc 0.808, time 51.2 sec\n",
      "epoch 5, loss 0.0006, train acc 0.945, test acc 0.885, time 51.2 sec\n"
     ]
    }
   ],
   "source": [
    "lr, num_epochs, batch_size, device = 0.05, 5, 256, d2l.try_gpu()\n",
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear or type(m) == nn.Conv2d:\n",
    "        torch.nn.init.xavier_uniform_(m.weight)\n",
    "net.apply(init_weights)\n",
    "criterion = nn.CrossEntropyLoss()\n",
    "train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)\n",
    "d2l.train_ch5(net, train_iter, test_iter, criterion, num_epochs, batch_size, device, lr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "6-n8R4cmc3kG"
   },
   "source": [
    "## Summary "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "Y4k_2Gwkc3kI"
   },
   "source": [
    "• Residual blocks allow for a parametrization relative to the identity function $f (x) = x$.\n",
    "\n",
    "• Adding residual blocks increases the function complexity in a well-defined manner.\n",
    "\n",
    "• We can train an effective deep neural network by having residual blocks pass through cross-layer data\n",
    "channels.\n",
    "\n",
    "• ResNet had a major influence on the design of subsequent deep neural networks, both for convolutional\n",
    "and sequential nature."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "3npW-MpDc3kJ"
   },
   "source": [
    "## Exercises "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "RhGq7r4zc3kL"
   },
   "source": [
    "1. Refer to Table 1 in the $ResNet paper^{147}$ to implement different variants.\n",
    "2. For deeper networks, ResNet introduces a “bottleneck” architecture to reduce model complexity. Try\n",
    "to implement it.\n",
    "3. In subsequent versions of ResNet, the author changed the “convolution, batch normalization, and\n",
    "activation” architecture to the “batch normalization, activation, and convolution” architecture. Make\n",
    "this improvement yourself. See Figure 1 in He et al., $2016^{148}$ for details.\n",
    "4. Prove that if x is generated by a ReLU, the ResNet block does indeed include the identity function.\n",
    "5. Why can’t we just increase the complexity of functions without bound, even if the function classes are\n",
    "nested?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "Hh1QqpPhc3kO"
   },
   "source": [
    "## References "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "w7t595OZc3kQ"
   },
   "source": [
    "[1] He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In Proceedings\n",
    "of the IEEE conference on computer vision and pattern recognition (pp. 770-778).\n",
    "\n",
    "[2] He, K., Zhang, X., Ren, S., & Sun, J. (2016, October). Identity mappings in deep residual networks. In\n",
    "European Conference on Computer Vision (pp. 630-645). Springer, Cham."
   ]
  }
 ],
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